Your reservation should say at the bottom (on the web, click on checked bags) how many bags you will be able to checkin outbound and inbound and cost. If it shows $0 for two, keep a printout.

I was starting my journey on a partner in Economy and then United on Polaris back in October. They asked me to pay first but when I pointed out, they were fine. Yours sound like all UA and domestic, but posting this anyway to help in future searches.